Geophysicists and associated professionals desire regular or regularized data from subsurface geological surveys, such as seismic surveys, for use in geophysical applications in order to accurately produce images of subsurface geology. Examples of such geophysical applications that require regularized data may include amplitude analysis in offset domain, seismic migrations, and merging of various 3D subterranean surveys to gain visual profiles of subterranean formations. However, during data acquisition, acquisition geometries, equipment failures, economical limitations and field obstacles, such as caverns, rivers, buildings, etc. often result in collection of irregular data, which may include blanks or gaps (i.e. non-collection of regular data). FIG. 1 is an exemplary example to show the data gaps in a typical onshore field survey. Such gaps and irregular data may adversely affect construction of subsurface geological structures to be used in geophysical explorations.
Data interpolation may typically be performed to fill in gaps in the irregular data in an attempt to produce regular data. There are a number of interpolation algorithms available in the industry. Fourier-based interpolation algorithm is one of the most versatile approaches. It is relatively computationally fast, and easily extends to higher dimensions to obtain an optimal data reconstruction. Minimum Weighted Norm Interpolation (MWNI) belongs to this family of Fourier-based interpolation algorithms. Although MWNI is currently used in processing seismic surveys to interpolate for missing data and to produce regular data from otherwise irregular data, one fundamental limitation of an MWNI algorithm is that it cannot properly process spatially aliased data. One approach to address the issue of aliased data is to apply filtering in removing the aliased energy before the application MWNI interpolation. The filtering ensures that the data are not spatially aliased, but unfortunately it also degrades the interpolation result, especially causing the poor reconstruction of steeply dipping subsurface structures. Another approach to minimize the data aliasing issue involves two steps. The first step uses MWNI to interpolate missing data in the frequency ranges that are not spatially aliased. The second step uses predictor filters to interpolate the missing data in frequency ranges that are spatially aliased. The large data gaps and highly irregular data often cause the construction of the prediction filters to fail. The industry-standard technique to handle the data aliasing issue in MWNI employs a bootstrapping method to uses a lower-frequency solution to constrain a higher-frequency solution. This approach assumes that low frequency signals are unaliased and existed in the data. However, typical seismic acquisitions have difficulties to record low-frequency signals, particularly in the frequency range between 1 to 6 Hz. In addition, the higher-frequency solution becomes aliased when that frequency reaches into the aliased frequency range. The use of a lower-frequency solution to constrain a higher-frequency solution does not resolve the data aliased issue. At the present time, there is no viable option to overcome the aliased issue in the MWNI method.
What is needed is a method that addresses shortcomings related to use of an MWNI algorithm to handle spatial aliased data.